| L(s) = 1 | + 2.11·2-s − 3.26·3-s + 2.46·4-s − 6.89·6-s − 1.21·7-s + 0.971·8-s + 7.64·9-s − 0.657·11-s − 8.02·12-s − 3.70·13-s − 2.57·14-s − 2.86·16-s + 4.59·17-s + 16.1·18-s + 5.68·19-s + 3.97·21-s − 1.38·22-s − 8.39·23-s − 3.17·24-s − 7.81·26-s − 15.1·27-s − 2.99·28-s + 9.06·29-s + 4.34·31-s − 8.00·32-s + 2.14·33-s + 9.70·34-s + ⋯ |
| L(s) = 1 | + 1.49·2-s − 1.88·3-s + 1.23·4-s − 2.81·6-s − 0.460·7-s + 0.343·8-s + 2.54·9-s − 0.198·11-s − 2.31·12-s − 1.02·13-s − 0.688·14-s − 0.716·16-s + 1.11·17-s + 3.80·18-s + 1.30·19-s + 0.868·21-s − 0.295·22-s − 1.75·23-s − 0.647·24-s − 1.53·26-s − 2.92·27-s − 0.566·28-s + 1.68·29-s + 0.779·31-s − 1.41·32-s + 0.373·33-s + 1.66·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 2.11T + 2T^{2} \) |
| 3 | \( 1 + 3.26T + 3T^{2} \) |
| 7 | \( 1 + 1.21T + 7T^{2} \) |
| 11 | \( 1 + 0.657T + 11T^{2} \) |
| 13 | \( 1 + 3.70T + 13T^{2} \) |
| 17 | \( 1 - 4.59T + 17T^{2} \) |
| 19 | \( 1 - 5.68T + 19T^{2} \) |
| 23 | \( 1 + 8.39T + 23T^{2} \) |
| 29 | \( 1 - 9.06T + 29T^{2} \) |
| 31 | \( 1 - 4.34T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 1.50T + 41T^{2} \) |
| 43 | \( 1 + 6.16T + 43T^{2} \) |
| 47 | \( 1 + 0.492T + 47T^{2} \) |
| 53 | \( 1 - 1.85T + 53T^{2} \) |
| 59 | \( 1 + 5.83T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 + 9.03T + 67T^{2} \) |
| 71 | \( 1 - 7.93T + 71T^{2} \) |
| 73 | \( 1 - 9.57T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 + 3.31T + 83T^{2} \) |
| 89 | \( 1 + 4.76T + 89T^{2} \) |
| 97 | \( 1 + 6.06T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.36243395495483213412347884820, −6.55069352378472114171642808774, −6.02636543459348960084729312966, −5.62351720159448669217740670418, −4.72414954060062142366757886795, −4.56885561205493412745680569719, −3.47396696755375370025319853492, −2.61715193372416955508664034408, −1.22405003773845350713734009556, 0,
1.22405003773845350713734009556, 2.61715193372416955508664034408, 3.47396696755375370025319853492, 4.56885561205493412745680569719, 4.72414954060062142366757886795, 5.62351720159448669217740670418, 6.02636543459348960084729312966, 6.55069352378472114171642808774, 7.36243395495483213412347884820
